A Recursive Construction for Skew Hadamard Difference Sets

Abstract

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.

Pure Gauss sums and skew Hadamard difference sets

“…After their work, there have been many studies on constructions and classification of skew Hadamard difference sets. See, e.g., a short survey in Introduction of [19]. In particular, Feng and Xiang [9] gave a construction of skew Hadamard difference sets based on pure Gauss sums, which are also in index 2 case.…”

“…In particular, Aoki (1997) classified pure Gauss sums for extension degrees f = 1, 2, 3, 4. In this paper, as a continuous study, we further characterize pure Gauss sums for odd extension degrees and classify them for f = 5, 7,9,11,13,17,19,23. Furthermore, we characterize a special subclass of pure Gauss sums in view of an application for skew Hadamard difference sets.…”